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A partial order <= on a SeT X is a MathematicalRelation satisfying the following conditions: for every a,b,c in X, a <= a (reflexive property) If a <= b, b <= a, then a = b (antisymmetric property) If a <= b, b <= c, then a <= c (transitive property) A set with a partial order on it is called a partial ordered set, or poset. A poset where any two elements have both a greatest lower bound and a least upper bound forms an algebraic structure called a LatticE. Every poset (X,<=) has a unique dual poset (X,>=). Given any a,b in X, we define the OpenInterval? (a,b) = {x in X : a < x < b}. Arbitrary unions of OpenInterval?s define a TopOlogy on the poset, called the OrderTopology?. Most familiar topologies are built up from objects of this sort, e.g. the RealNumbers?. Examples of partial orders include implications and inclusions ("is a subset of" and the more general "is a subobject of"). |
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#REDIRECT Partial order |
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